Graph Topologies and Uniform Convergence in Quasi-Uniform ...

More documents

Recommendations

Info

136 j. rodríguez-lópez, s. romagueraNow, we present some examples where we can apply this result.Example 1. As we observed at the introduction, if f : (X, T ) → (Y, T ′ )is a continuous function and U denotes the Pervin (resp. point finite, locallyfinite, semi-continuous, fine transitive, fine) quasi-uniformity on X (see [8] forthe corresponding definitions) and V the corresponding quasi-uniformity onY then f : (X, U) → (Y, V) is quasi-uniformly continuous (see [8, Proposition2.17]). Therefore, we deduce that T (H U −1 ×V) = T (V X ) on C(X, Y ).Now, we recall some definitions about the theory of lattices. Our mainreference is [9].A partially ordered set (or a poset) is a nonempty set L with a reflexive,transitive and antisymmetric relation ≤. A lattice is a poset where everynonempty finite subset has infimum and supremum.Given a lattice L and O ⊆ L, we denote ↑ O = {l ∈ L : there exists o ∈O such that o ≤ l}.A lattice is said to be complete if every nonempty subset has infimum andsupremum.Let L be a complete lattice. We say that x is way below y, and we denoteit by x ≪ y, if for all directed subsets D of L the relation y ≤ sup D impliesthe existence of d ∈ D such that x ≤ d, where D is called directed set if givenp, q ∈ D we can find l ∈ D such that p ≤ l and q ≤ l.A continuous lattice is a complete lattice L such that it satisfies the axiomof approximation:x = sup{u ∈ L : u ≪ x}for all x ∈ L.Let L be a complete lattice. The Scott topology σ(L) is formed by allsubsets O of L which satisfy:i) O =↑ Oii) sup D ∈ O implies D ∩ O ≠ ∅ for all directed sets D ⊆ L.We recall that a semigroup is a pair (X, ·) such that · is an associativeinternal law or operation on X for which exists an identity element e.Definition 3. If (X, T ) is a topological space and (X, ·) is a semigroupsuch that the function ·x : X → X defined by ·x(y) = x · y for all y ∈ X iscontinuous for all x ∈ X we say that (X, ·, T ) is a semitopological semigroup.
graph topologies and quasi-uniform convergence 137Definition 4. A Scott quasi-uniform semigroup is a pair (L, U) where Lis a complete lattice, U is a quasi-uniformity on L such that T (U) is the Scotttopology and (L, ∨, T (U)) is a semitopological semigroup.Now, we prove our main result.Theorem 1. Let (X, U) be a quasi-uniform space and (L, V) a Scottquasi-uniform semigroup. The following statements are equivalent:i) Every continuous function from X to L is quasi-uniformly continuous.ii) The proximal topology of (X × L, U −1 × V) agrees with the topology ofuniform convergence on C(X, L).iii) The upper Hausdorff quasi-uniform topology induced by U −1 × V agreeswith the topology of uniform convergence on C(X, L).Proof. i) ⇒ ii) This is deduced from Proposition 3.ii) ⇒ iii) We only have to use Proposition 1.iii) ⇒ i) Suppose that there is a continuous function f : (X, T (U)) →(L, σ(L)) which is not quasi-uniformly continuous. Then, we can findW ∈ V and two nets {a U } U∈U and {b U } U∈U such that (a U , b U ) ∈ U and(f(a U ), f(b U )) ∉ W for all U ∈ U. For each U ∈ U we define the followingfunction f U : X → L:⎧⎨f(b U ) if x ∈ {a U }f U (x) =⎩f(x) ∨ f(b U ) otherwise.Let us prove that f U is continuous for each U ∈ U. Let us fix U ∈ U andlet x ∈ X and {x λ } λ∈Λ be a T (U)-convergent net to x. We distinguish thefollowing cases:Case 1. x ∈ {a U }We have that f U (x) = f(b U ), and the values of f U (x λ ) can be f(b U ) orf(x λ ) ∨ f(b U ) for all λ ∈ Λ. It is evident that in both cases we obtain thatf U (x) ≤ f U (x λ ), so f U is a continuous function in x.Case 2. x ∈ X\{a U }Since {x λ } λ∈Λ is T (U)-convergent to x, it is eventually in X\{a U }. Therefore,by continuity of the sup operation, we deduce that f U is continuous in x.Consequently, we have shown that f U is continuous for all U ∈ U.We now prove that {f U } U∈U converges to f with respect to T (H + U −1 ×V ).
Page 3 and 4: graph topologies and quasi-uniform
Page 5: graph topologies and quasi-uniform
Page 9 and 10: graph topologies and quasi-uniform

Graph Topologies and Uniform Convergence in Quasi-Uniform ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?